Given a spherical fibration over the classifying space of a finite group we define a dimension function for the –fold fiber join of , where is some large positive integer. We show that the dimension functions satisfy the Borel–Smith conditions when is large enough. As an application we prove that there exists no spherical fibration over the classifying space of with –effective Euler class, generalizing a result of Ünlü (2004) about group actions on finite complexes homotopy equivalent to a sphere. We have been informed that this result will also appear in upcoming work of Alejandro Adem and Jesper Grodal as a corollary of a previously announced program on homotopy group actions due to Grodal.
"Dimension functions for spherical fibrations." Algebr. Geom. Topol. 18 (7) 3907 - 3941, 2018. https://doi.org/10.2140/agt.2018.18.3907